ank exists for each ]c (in which case

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1 BANACH ALGEBRA AND SUMMABILITY BY ALBERT WILANSKY AND KARL ZELLER 1 Several Banach algebras arise naturally in summability theory These are algebras of infinite matrices A (ak), n, ] 1, 2, The study of these algebras may have two products: the generation of examples of Banach algebras with certain properties; and, on the other hand, applications to the theory of summability which has, since 1929, benefited greatly from the theory of Banach space and linear topological space in general An example of such an application of Banach algebra is the observation that an element near the identity element has an inverse, which was applied to summability by Agnew and, for instance, was used extensively in [7] 2 Let AII supk ak I Then if A <, A is an endomorphism (= continuous linear transformation) of m, the Banach space of all bounded complex sequences, and A is its norm However, much significance is attached to the way in which A transforms sequences not in m Let, be the set of all matrices A with AII < Then is a Banach algebra;it is a closed proper subalgebra of the algebra of all endomorphisms of m Let c, Co be the spaces of convergent and null sequences For each n, ]c, [ak --< A and so the entries are continuous linear functionals on Let T be the subset of consisting of the triangular matrices, ie the matrices A such that auk 0 if ]C > n Since the entries are continuous, T is closed in It is clearly a subalgebra and has its inverses, ie A e T, A-le together imply A-le T Let F be the subset, clearly a subalgebra, of I, consisting of the conservative matrices, ie matrices A such that Ax e c whenever x e c A matrix A e belongs to F if and only if ak lim ank exists for each ]c (in which case [ ak < oo) and lim k ak exists Since auk and k ak are all linear functionals of unit norm, this shows that F is closed in We have also, [8], that F contains its inverses, ie A e P, A -1 e imply A- F It is in the proof of this result that we are forced (as far as we know) to consider the action of A on unbounded sequences For A P, let x(a) lim k ank ak Then x is a multiplicative linear functional, [8] If x(a) 0, A is called co-null, otherwise co-regular These terms were introduced in [6] Let A P n T, the set of conservative triangular matrices Many of the classical summability methods are defined by matrices in A, for example the Cesro, Nhrlund and Hausdorff methods We shall see, below, several fundamental structural differences between A and P, and individual differences e Received June 17, 1957 The second author holds a National Science Foundation research grant 378

2 BANACH ALG]BRA AND SUMMABILITY 379 between their members For example, if A e A and A-le A, then A must be a triangle (ie triangular and a 0 for each n) In this case A sums no divergent sequences, ie Ax c if x e c For bounded x this is trivial since A and A -1 preserve c as operators on re;for unbounded x it is also true since A-I(Ax) x The situation is quite different for F If A e F and A-le F, A sums no bounded divergent sequences, but may sum unbounded ones Indeed A need not be 1-1 for sequences outside m; see [8], Theorem 7, for a regular matrix which is its own inverse but is not 1-1; also we may consider a "principal diagonal matrix"; see [7] The crucial point is contained in this remark" Remark For A, B e T and any sequence x, Bx exists and A(Bx) AB(x), This remains true with instead of T if x is assumed bounded, but not if x is not assumed bounded Indeed there exists A F with A -1 e and sequence x 0 such that Ax 0, as was pointed out just preceding the remark 3 Let us consider an algebra with identity I Following [5] we define the radical to be the set of points A such that I BA has an inverse for every B (In our case, I BA will occasionally have an inverse matrix which does not lie in the space considered, ie no inverse "exists") We note the facts, given in [5]:I BA has an inverse if and only if I AB does The radical is a two-sided ideal If the algebra is a Banach algebra, the radical is closed The algebra is called semisimple if the radical contains only 0 The spectrum of A is the set of scalars z such that A zi has no inverse If A is in the radical, its spectrum contains only 0 It is clear that if f is a linear and multiplicative functional, f O, then for each A, f(a) is in the spectrum of A (since f(a IfA) 0 and fz is an ideal) Hence every element A in the radical has f(a) 0 This, applied to A with f x, proves that every matrix in the radical of A is co-null Theorem 2, below, proves much more than this We can also apply this remark to A or T with f(a) an, for a fixed n This proves that every matrix in the radical of z or T has all its diagonal elements zero THEOREM 1 and r are semisimple T and n are not semisimple Given A e, A 0, let x c, Ax O For example, if, for a certain n, k, we have ak 0, we may take x ti, a sequence of zeros save for a one in the ]ct place Define B e F as follows: B consists entirely of zeros except for a single column, and BAx x Then I BA carries x into 0, hence is not 1-1 and has no inverse This proves that A is not in the radical of or F, and hence that these spces are semisimple To prove the second part of the theorem is relatively easy For example, a triangular matrix with a single nonzero element, and that one occurring below the main diagonal, is easily seen to be in the radical of T and A However, we can give an exact description of these radicals

3 380 ALBERT WILANSKY AND KARL ZELLER THEOREM 2 For both T and A, a matrix A is in the radical if and only if its diagonal elements are zero and kl ankl is uniformly convergent For A this last condition is equivalent to "A is coercive," ie x is bounded Ax is convergent whenever The proof of the "if" part of this theorem was suggested by J A Schatz The details of the proof were worked out by E K Dorff We proved just before the statement of Theorem 1, that a matrix in the radical of A or T has zero diagonal Next, we shall show that a radical matrix A must have the uniform convergence property mentioned Consider first the case in which A is conservative, ie A e A Suppose that k ak is not uniformly convergent We first observe that A is not coercive since a coercive matrix must have, along with other properties, uniform convergence of the series mentioned Thus there exists a bounded sequence x such that y Ax is divergent; y must be bounded since A has finite norm, hence y has two distinct limit points, say yp(,) --+ a, yq(n) --+ We may assume p(n) T o q(n) each n Let B be defined as follows" for n < q(1) and all ], let b 0; for q(r) <= n < q(r q- 1) let b, 0 for ] p(r), ]c q(r), while Xn Yq(r) bn,q(r) yp(r) yq(r) y,(r) yq(r) The matrix B belongs to where M is an upper bound for all x I, [Y [; b 1 for each n >_- q(1), and each column of B terminates in a string of zeros (Thus, in fact, B is regular) Also By is a sequence whose n th term, for n >- q(1) is x Hence (I BA)x x By is a sequence terminating in zeros, a convergent sequence; call it z Thus the matrix I BA is a conservative triangular matrix which transforms a divergent sequence, x, into a convergent one, z Thus I BA cannot have an inverse, D, in A, since Dz would have to be convergent, while Dz x Hence A is not in the radical of A (The above construction is inspired by [1]) In the second case where A is not necessarily conservative, let us assume that A is in the radical and that 7 an is not uniformly convergent We -m(+i) can find fori 1, 2, LetBbea diagonal matrix withbnn lforn n(1), n(2), b 0 otherwise Then BA, a row submatrix of A, has no row submatrix with the uniform convergence property Similarly, form C such that CBA is a row submatrix of BA with the sequence of row sums convergent Then form DCBA with convergent first column, EDCBA with convergent second column, and so on Each of B, C, D, E, is diagonal with zeros and ones on the diagonal Let F be the matrix whose first row is the first row of A, whose second row is the second row of BA, whose third row is the third row of CBA, and so on; F is a submatrix of A, thus F GA with G Yp(r) Xn

4 BANACtt ALGEBRA AND SUMMABILITY 381 diagonal with zeros and ones on the diagonal; F is conservative but not coercive As above we can find H e A such that I HF has no inverse in A Then it has no inverse in T for F contains its inverses in (I, and so A contains its inverses in T Thus F is not in the radical of T But F GA, and this contradicts the assumption that A is in the radical and the fact that the radical is an ideal We now turn to the proof that if k a is uniformly convergent nd 0 for each n, then A belongs to the rdicm of T or A, whichever is if a appropriate First let us notice that A is u limit point of right-finite matrices, ie matrices B with b 0 for/c > m, m being some number depending only on B (This is not to be confused with row-finite, in which m depends on n) This is true since given e > 0 we choose K such that k>k ank < e for all n Let bk a for ]c < K, b 0 otherwise Then B is right-finite and lib- d]] < Since the rdiem is closed, we have only to prove tha any right-finite mtrix with zero diagonal is in the radical Since the radicm is closed under addition, it is sufficient to prove that matrix with only one nonzero column and zero diagonal is in the radical This we now do Let m be positive integer, and let a 0 for M1 n, k excep when k m; leta 0 for n _-< m Let B be an arbitrrymatrix in T (If A A, tak B A) Let D I BA exists Our next step is to see that D- < Let x be an rbitrary bounded sequence, and y D-x Thenx Dy y-- BAy; hence Yn Xn + Ym bnrarm aiid [yn txn[ + [Ym[" I!ali ][A r-----m+l Thus y is bounded, and D -1 is a matrix which preserves boundedness, ie ]]D-1]I < HenceD -l e w and so A is in the radical of w In caseaea, Clearly d 1 and D is triangular; heno D- then D e A and D -1 e T; as mentioned in 2 then D -1 e A Hence A is in the radical of A It is perhaps of interest to observe that the matrices in the radical are totally continuous operators on m since they are limits of operators of finite rank 4 Contained in the proof of Theorem 2 is the following result: LEMMA 41 In T, if A has an inverse (in T), then so does A + B for any right-finite B e T such that A + B is a triangle Related to this is e LEMMA 42 Let A e A and] ann l for all n, and let 1 ak > [[ A ]] 2 A- Then A, and A sums no divergent sequences The result is best possible in that the condition a 1 cannot be dropped and the 2 cannot be increased m--1 Proof Let _k=, [ak > > A 2; then for lrgen, ak > t, and so 2 a -_ ]a 1 :t a < A 1 < 1, -

5 382 ALBERT WILANSKY AND KARL ZELLER and A differs by a right-finite matrix from a matrix in the unit neighborhood of I By Lemma 41, the result follows To see that it cannot be improved in the way mentioned, consider first the Cesro matrix It satisfies ]akl > I]AII 2 but not a 1 Consider also the matrixa with ann an,n-1 1, ak 0 otherwise This matrix has a 1 and The result can be improved by replacing a 1 by a condition of the type lim a 1 with appropriate modifications An application of Lemma 42 is the known result that x is convergent if lx + -1 a xk is convergent, l ak < for the corresponding matrix A has lal ]IAI] 1 5 The set of co-null matrices is a two-sided ideal in, and a maximal linear subspace of 1 (since x is linear and multiplicative) but is not, clearly, an ideal in Indeed there exists co-null A with right inverse in (it could not have a left inverse in ); see [8] This shows that x cannot be extended to (I, so as to be linear and multiplicative We examine other subsets of P for the property of being an ideal For any sequence x, let (x) be the set of conservative matrices which sum x, ie transform x into a convergent sequence Clearly (x) F if and only if x is convergent, or equivalently I (x) if and only if (x) P LEMMA 51 (X) C (y) if and only if (x) (y) or (y) F This result is essentially due to Brudno, and t Erd6s and Rosenbloom, who prove it for bounded x; see [4] If x is unbounded, choose A e r such that A transforms into convergent sequences only sequences of the form ax -t- u, where a is a scalar, and u is convergent See [7] for existence of A Since A e (y), it follows that y a x + u THEOREM 3 In F, (x) is a right ideal if and only if x is convergent, a left ideal if and only if x is bounded (x) n A is, in A, a right ideal if and only if x is convergent, but is always a left ideal If x is bounded, we simply consider it as a point of m, and since we are dealing with endomorphisms of m, the result is trivial since A(Bx) (AB)x for any A, B e The same is true for A, B e A, whether x is bounded or not since A is row-finite See the earlier remark Next we assume that x is unbounded and show that (x) is not a left ideal in F There exists, [7], A (x) such that A has a two-sided inverse, iu fact IIA Ill < 1 Next we assume that x is divergent and show that (x) is not a right ideal Buck [2, 3] has shown that if A is any regular matrix and x is divergent, there exists a subsequence of x not summable by A Let B be the matrix such that Bx is the subsequence of x; then AB (x) and the result follows

6 BANACH ALGEBRA AND SUMMABILITY 383 LEMMA 52 Let x be such that (x) is a closed subset of F (x) n A is a closed subset of /), then f(a) lim Ax for A (x) defines a continuous function f o (x) (o (x) a) Here lim Ax means the limit of the convergent sequence Ax The result follows from the Banach-Steinhaus closure theorem and the facts f(a) lim limm ak xk, 7 a x <_- max k=l k=l l_k_m THEOREM 4 (X) is closed in F if and only if x is bounded (x) A is closed in A if and only if x is bounded Let x be bounded Let un(a) =1 ark xk Then lug(a)] <= (sup x I)ll A Thus/u is a uniformly bounded sequence of functionals, and so (x) is closed, being the set of all A such that lim u(a) exists For x unbounded, let r be any number, and n a subscript such that x > r Let A A be a matrix with only one nonzero column, the nh, which has n zeros and then ones Then A 1, lira Ax ]x > r By Lemma 52, (x) is not closed This completes the proof We could have shown that (x) is not closed in F for unbounded x by noting that I e (x); but this is false for A since I AII < 1, A e A, implies that A sums no divergent sequences Since we can, in fact, find a regular matrix A e (x) with III A as small as we wish, this yields a similar result for the subset of F consisting of multiplicative matrices, ie such that ak 0 for all It (They are called multiplierrive since for x c, Ax converges to a constant multiple, viz lim a, of lim x) But this is not so for the subset of consisting of multiplicative matrices; call this subset M THEOREM 4 (X) M is closed in M if and only if x is bounded Let x be unbounded As for the previous result we need merely show that given any positive integer s, there exists A e (x) i with AI] 1, lim Ax s (This cannot be found in [1], although it contains many examples of this type) Let {Xpn be a subsequence of x with xp > max (s, n) For n < p let auk n (the Kronecker delta), for p =< n < p+, let a 0 for lc p, ak S/Xpr if / p 5 Agnew, [1], has given several results of this type: if x and y are sequences, x divergent, then there exists a regular matrix A with y Ax He remarks that A cannot be expected to be triangular In fact, as we shall now see, the situation is worse with respect to triangular matrices, in that we cannot even expect to come close to y with Ax This is clearly related to the semi-

7 384 ALBERT WILANSKY AND KARL ZELLER simplicity of q) and lack thereof of T; for if, given arbitrary x, y, we could find A e T with Ax y, then, given B e T, let x By, choose A with Ax y, and so (I AB)y O, and B is not in the radical of T; hence T would be semisimple THEOREM 5 Let x be a bounded divergent sequence, y a bounded sequence; then there exists A e A such that y Ax has a finite number of nonzero terms, and hence is convergent There exist unbounded sequences x, y such that y Ax is divergent for every A e A The matrix mentioned in the first part of the theorem is the one called B, constructed in the proof of Theorem 2 so that By has x for its n h term for n sufficiently large For our present purpose we have interchanged x, y For the second part of the theorem, let B be the matrix given by bn-1 e, n 1, 2,- ;bn 0for]c n 1;where lenl < Letybea sequence such that x By is unbounded By Theorem 2, B is in the radical of A; hence, for any A A, I AB has an inverse and so does not transform any divergent sequence into c In particular (I AB)y c, and the theorem is proved Let s be the set of all sequences Agnew s results, mentioned above, state that the map from I to s given by A --- Ax is onto, x being a fixed unbounded sequence Indeed it is onto from the subset of I consisting of the regular matrices Theorem 5 says that it is onto sic as a map of A if x is bounded and divergent, but need not be onto sic if x is unbounded 7 The maximal group G of a Banach algebra is the set of elements with inverses The elements which are in the closure of the maximal group are of some general interest Mercer s theorem states that the Cesro matrix is such an element The following example is instructive Let A be the multiplicative matrix given by ann 1, a s, a 0 otherwise Then for0 =< s < 1, IlI- A[[ s < 1;henceAA, andaeg Fors > 1, let B be the matrix gotten from A by omitting the first row Then B e F, I s-lb 1Is < 1; hence B -1 e F, and, [7], Bx convergent implies that x u -t- av, where u e c, a is a scalar, and v is a fixed unbounded sequence (Here we also have used the fact that the set of sequences v such that By 0 is one-dimensional) But Bx is convergent if and only if Ax is Hence for 0 =< s < 1 and s > 1, A sums no bounded divergent sequences; and in the first case, 0 =< s < 1, A belongs to the maximal group The dividing line s 1 yields the matrix A1 which sums the bounded divergent sequence {(-1)} We conjecture that this behaviour is general" ie, let A be a matrix in A with ann 0 and with A not in the maximal group (ie A sums divergent sequences); then (we conjecture) A sums bounded divergent sequences if and only if it is in the closure of the maximal group Some writers have classed as pathological those matrices not summing any bounded divergent sequences The conjecture would relegate the nonpathological

8 BANACH ALGEBRA AND SUMMABILITY 385 matrices to a thin veneer on the maximal group As an indication of the possible difficulty of the problem we note the proof of Mercer s theorem REFERENCES 1 R P AGNEW, On ranges of inconsistency of regular Sransformations, and allied topics, Ann of Math (2), vol 32 (1931), pp R C BUCK, A note on subsequences, Bull/kmer Math Soc, vol 49 (1943), pp , An addendum to "A note on subsequences", Proc Amer Math Soc, vol 7 (1956), pp ) EIDSS AND P C ROSENBLOOM, Toeplitz methods which sum a given sequence, Bull Amer Math Soc, vol 52 (1946), pp E HILLE, Functional analysis and semi-groups, Amer Math Soc Colloquium Publications, vol 31 (1948) 6 A WlLANSXY, An applicalion of Banach linear functionals to summability, Trans Amer Math Soc, vol 67 (1949), pp A WILANSKY AND K ZELLER, Summalion of bounded divergent sequences, topological methods, Trans Amer Math Soc, vol 78 (1955), pp The inverse matrix in summabilily: reversible matrices, J London Math Soc, vol 32 (1957), pp LEHIGH UNIVERSITY BETHLEHEM PENNSYLVANIA WAYNE STATE UNIVERSITY DETROIT MICHIGAN